Chapter 5: Recap

Directed acyclic graphs

Directed acyclic graphs are the brain child of computer scientist Judea Pearl, who developed them to reason systemically about causal inference. Judea Pearl’s The Book of Why provides an accessible introduction to directed acyclic graphs and makes for a nice holiday reading.

DAGs are build from two types of components:

• Nodes, which represent variables
• Edges, which represent relationships between variables

All graphs are made of nodes and edges. What is special about directed acyclic graphs is that

1. the relationship between two variables has a direction, such that
   • variables at the start of the arrows are causes and
   • variables at the end are effects
2. a succession of arrows must not form a circle, that is, a variable cannot cause itself, even indirectly¹. For example, \(\small D\rightarrow B\rightarrow C\rightarrow D\) is not allowed.

Confounding bias: Common cause of exposure and outcome

• B = brain volume
• D = disorder
• S = sex

C_dag = dagitty(
  "dag{
  S->B;
  S->D;
  B->D
  }")
coord.list = 
  list(
    x=c(B=0,D=2,S=1),
    y=c(B=0,D=0,S=-1))
coordinates(C_dag) = coord.list
drawmydag(C_dag, cex = 1, lwd = 1)

C_dag_A = C_dag
adjustedNodes(C_dag_A) = "S"
drawmydag(C_dag_A, cex = 1, lwd = 1)

Confounding bias	Adjustment

Bias from adjustment of mediators

• T = treatment
• M = motivation
• W = well being

M_dag = dagitty(
  "dag{
  M->W;
  T->M;
  T->W
  }")
coord.list = 
  list(
    x=c(T=0,W=2,M=1),
    y=c(T=0,W=0,M=-1))
coordinates(M_dag) = coord.list
drawmydag(M_dag, cex = 1, lwd = 1)

M_dag_A = M_dag
adjustedNodes(M_dag_A) = "M"
drawmydag(M_dag_A, cex = 1, lwd = 1)

Mediation by M	Bias from adjustment of mediator

Bias from adjustment of colliders

• E = internalising
• I = externalising
• B = in BUP

Cl_dag = dagitty(
  "dag{
  I->B;
  E->B;
  E->I
  }")
coord.list = 
  list(
    x=c(E=0,I=2,B=1),
    y=c(E=0,I=0,B=1))
coordinates(Cl_dag) = coord.list
drawmydag(Cl_dag, cex = 1, lwd = 1)

Cl_A_dag = Cl_dag
adjustedNodes(Cl_A_dag) = "B"
drawmydag(Cl_A_dag, cex = 1, lwd = 1)

Collider	Selection bias from conditioning on a collider

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1. If one has multiple measurements of the same variable over time, each measurement would be a new node↩︎